To find the missing justification, we need to determine how the coordinates of points [tex]\( D \)[/tex] and [tex]\( E \)[/tex] are calculated. Here’s the step-by-step solution:
1. Identify the midpoints:
- Let's assume point [tex]\( A \)[/tex] has coordinates [tex]\((2, 2)\)[/tex] and point [tex]\( B \)[/tex] has coordinates [tex]\((6, 8)\)[/tex].
- The midpoint [tex]\( D \)[/tex] of segment [tex]\( AB \)[/tex] is calculated using the midpoint formula:
[tex]\[
D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
- Substituting [tex]\((x_1, y_1) = (2, 2)\)[/tex] and [tex]\((x_2, y_2) = (6, 8)\)[/tex]:
[tex]\[
D = \left( \frac{2 + 6}{2}, \frac{2 + 8}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5)
\][/tex]
- Now, let's assume point [tex]\( C \)[/tex] has coordinates [tex]\((8, 4)\)[/tex].
- The midpoint [tex]\( E \)[/tex] of segment [tex]\( BC \)[/tex] is calculated similarly:
[tex]\[
E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right)
\][/tex]
- Substituting [tex]\((x_2, y_2) = (6, 8)\)[/tex] and [tex]\((x_3, y_3) = (8, 4)\)[/tex]:
[tex]\[
E = \left( \frac{6 + 8}{2}, \frac{8 + 4}{2} \right) = \left( \frac{14}{2}, \frac{12}{2} \right) = (7, 6)
\][/tex]
Therefore, the missing justification is:
The coordinates of point [tex]\( D \)[/tex] are [tex]\((4, 5)\)[/tex] and coordinates of point [tex]\( E \)[/tex] are [tex]\((7, 6)\)[/tex] because they are the midpoints of segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] respectively, using the midpoint formula.
